Dynamical Systems on Surfaces

208 Pages·1983·6.601 MB·English

by  Claude Godbillon (auth.)

Universitext
Advisors
F.W Gehring
P.R. Halmos
C.C. Moore
C.Godbillon
Dynamical Systems
on Surfaces
Translation from the French by
H. G. Helfenstein
With 70 Figures
Springer-Verlag
Berlin Heidelberg New York 1983
Claude Godbillon
Dllpartement de MatMmatiques, Universitll Louis Pasteur
7, rue Renll Descartes, F-67084 Strasbourg
Originai edition "Systemes dynamiques sur les surfaces"
© Strasbourg Lecture Notes
AMS Subject Classification (1980): 34 C, 58 F
ISBN·13: 978·3·540·11645·5 e·ISBN·13:978·3·642·68626·9
001: 10.1007/978·3·642·68626·9
Ubnuy of Congress Cataloging in Publleation Data. Godbillon, Claude, 1937-
Dynamical systems on surfaees. (Universitext). Translation of: Systemes dynamiques
aur les surfaees. Bibllography: p. 1. Differentiable dynamieal systems. 2. Follatlons
(Mathematlea) I. lltle. QA614.8.G6313. 1982. 516'.36. 83-19176
ISBN·13:978·3·540·11645·5 (U.S.)
This work is subject to copyright. All rights are reserved, whether the whole or part
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© by Springer-Verlag Berlin Heidelberg 1983
2141/3140-543210
Preface
These notes are an elaboration of the first
part of a course on foliations which I have given
at Strasbourg in 1976 and at Tunis in 1977.
They are concerned mostly with dynamical sys
tems in dimensions one and two, in particular with
a view to their applications to foliated manifolds.
An important chapter, however, is missing, which
would have been dealing with structural stability.
The publication of the French edition was re
alized by-the efforts of the secretariat and the
printing office of the Department of Mathematics
of Strasbourg. I am deeply grateful to all those
who contributed, in particular to Mme. Lambert for
typing the manuscript, and to Messrs. Bodo and Christ
for its reproduction.
Strasbourg, January 1979.
Table of Contents
I. VECTOR FIELDS ON MANIFOLDS
1. Integration of vector fields. 1
2. General theory of orbits. 13
3. Irlvariant and minimaI sets. 18
4. Limit sets. 21
5. Direction fields. 27
A. Vector fields and isotopies. 34
II. THE LOCAL BEHAVIOUR OF VECTOR FIELDS 39
1. Stability and conjugation. 39
2. Linear differential equations. 44
3. Linear differential equations with constant coefficients. 47
4. Linear differential equations with periodic coefficients. 50
5. Variation field of a vector field. 52
6. Behaviour near a singular point. 57
7. Behaviour near a periodic orbit. 59
A. Conjugation of contractions in R. 67
III. PLANAR VECTOR FIELDS 75
1. Limit sets in the plane. 75
2. Periodic orbits. 82
3. Singular points. 90
4. The Poincare index. 105
5. Planar direction fields. 116
6. Direction. fields on cylinders and Moebius stripso 123
A. Singular generic foliations of a disco 127
IV. DIRECTION FIELDS ON THE TORUS AND HOMEOMORPHISMS OF THE CIRCLE 130
1. Direction fields on the torus. 130
2. Direction fields on a Klein bottle. 137
3. Homeomorphisms of the circle without periodic point. 144
4. Rotation number of Poineare. 151
5. Conjugation of circle homeomorphisms to rotations. 159
A. Homeomorphism groups of an interval. 166
B. Homeomorphism groups of the circle. 170
V. VECTOR FIELDS ON SURFACES 178
1. Classification of compact surfaces. 178
2. Vector fields on surfaces. 181
3. The index theorem. 188
A. Elements of differential geometry of surfaces. 191
BIBLIOGRAPHY 200
Chapter I. Vector Fields on Manifolds
1. INTEGRATION OF VEeroR FIELDS
Let M be a differentiab1e manifo1d without boundary of dimen-
sion m and of e1ass eS 2.,,; s ~ + '" (respeetive1y ana1ytie) and 1et X
I I
be a veetor field on M of elas s er, 1 < r < s-l (respeetivelyanalytie).
1.1. DEFINlrION. An integral eurve of X is a map e of elass el of an
interval J of ~ into M satisfying e' (t) = X(e(t)) for all t ( J.
e
~
t
I
J
1. 2. Examp1es.
i) If y is a zero of X then the eonstant map of R onto y is
an integral eurve of X.
In this ease y is eal1ed a singular point of X. A point of M
where X does not vanish is a regular point.
ii) If e is an integral eurve of X, so is the map t<-+e(t+'f) I
for all 'f ( IR.
2
iii) If c is an integral curve of X the map t~ c (-t) is an integral
curve of the field -X.
iv) Let q: M--7M be a covering map. !'hen the tangent bundle T(M) is
isomorphic to the inverse image q*T(M) of the tangent bundle T(M) by
the projection q. Hence there is a uniquely defined vector field ~ on
M (in the same differentiability class as X) such that qT ~ = X • q.
0
e
In this case the projection c = qoc of an integral curve of
rv
X is an integral curve of X.
conversely every integral curve of X is the projectian of an inte-
,-.,
gral curve of X.
v) Let X and Y be vector fields on manifolds M and N respectively
and h: M~N a differentiable map satisfying hTo X = Yoh. Then the image
under h of any integral curve of X is an integral curve of Y.
1. 3. Remarks.
i) Let (Yl' ..• , y ) be a local system of coordinates on an open
m
set U of M, and let X be expressed on U as ?:ai _0_. Then the integral
~ a yi
curves of X in U are the solutions of the (autonomous) system of dif-
ferential eguations y1:. = a1·. (y 1 ' .... y m) , i = 1, •... m.
ii) If N = M ~ R, a vector field Y on N of the form Z(y,s) + __0 _
os
with Z(y,s) eT M, corresponds locally to a non-autonomous system of
y
differential equations y'. = b. (yl' ..• , Ym' t), i=l, ..• ,m.
1. 1.
The above remark (i) allows a reformulation of the local existence
and uniqueness theorems for solutions of differential equations, as
well as of the statements about their differentiable dependence on ini-
tial conditions as follows (cf. [lOJ, [12J):
1.4. THEOREM. For every point y of M and for every real number T there
3
exist an open neighhourhood U of y in M,
a number ~ > 0,
a map i of e1ass Cr (respeetive1y ana1ytie) of (T-E, T+E)X U
into M,
satisfying for every point x of U the following properties:
a) t~i{t,x) is an integra1 eurve of X7
b) i{T, 'x) X7
e) if e is an integra1 eurve of X defined on an interva1 eon-
taining T and sueh that e{T) = x, then e{t) = i{t,x) in a neighbour-
hood of T
Consequenees:
i) Two integra1 eurves of X interseeting in a point eoineide
in a neighbourhood of this point7
ii) 1et Ui' €~, ii' i = 1,2, be two sets of data as in theo
rem 1.4 with the properties a),b), and e). If ~ = inf{el ,Ez) and
V = u1n U2' then iti1= i2 on (T-E., T+~) X V.
1.5. COROLLARY. There exist an open neighbourhood W of tO} X M in
~ X M and a map m. of e1ass Cr (respeetive1y ana1ytie) of W into M with
the following properties satisfied at every point y of M:
a) IR X ty} n W is eonneeted7
b) t i{t,y) is an integra1 eurve of X7
-'10
e) i (O,y) = y7
d) if (t',y), (t+t',y) and (t,i{t',y» are in W, then
i{t+t' ,y)= i{t, Ht' "y».
Furthermore, if W., i., i=1,2, are two sueh data satisfying a),
l. l.
b), and e), then they a1so satisfy d), and i1= i2 on w1n W2.
4
Continuing with these notations we let V be an open set of M
such that "\t}X V and l-t\x q;(ltix V) are eontained in W. I'hen H{t}x V)
is open in M, and the map ~t: x~(t,x) is a diffeomorphism of V onto
this open set having the inverse CP-t: ZI-+q; (-t, z) •
In addition, for {t'} X V, {t+t'} X V, and It! X q;(lt'} X V) being
eontained in W as weIl, we have ~t+t'= CPtoCPt,on V. These remarks and the
considerations in §1.7 justify the following definition:
M
I
[II ----
1.6. DEFINITION. A local one-parameter group of diffeomorphisms (or a
flow) of elass cr(respeetively analytie) of M is an ordered pair (W,~),
where W is an open neighbourhood of loJx M in R X M and ~ a map of elass
Cr (analytie) of W into M, having at every point y of M the following
properties:
a)1R xty} n W is connected~
b) ~ (O,y) = y~
e) if (t' ,y), (t+t' ,y), and (t, ~ (t' ,y)) are in W then
f(t+t',y) = f (t, '(t',y)).
5
1.7. For W = RXM, ~ is a (global) one-parameter group of diffeomorph-
isms of M. In this case the map c!l : x~~ (t,x) is a diffeomorphism of M
t
for every tER , and we have:
i) C!lo = idencity map;
In other words, ~ is a differentiable (analytic) action of: R
on M.
Such a flow will often be denoted by (CPt)tER.
1.8. Remark. A vector field X on M allows the construction of a flow
(W,~) of diffeomorphisms of M (of the same differentiability class as X)
such that for every point y of M the curve t.~~(t,y) is an integral
curve of X. (W,~) is aloeal one-parameter group generated by X.
The germ of such a flow along to~x M is uniquely determined
by X (corollary 1.5).
Conversely, if (W,~) is a flow of class Cr (respectively ana-
lytic) on M, there exists one and onlyone generating vector field X
of class cr- l (respectively analytic): the value of X at a point yEM
is the vector tangent in y to the curve tl~~(t,y). This field whose
va 1u e at t he po~· nt y ~. s -a-. (O ,y ) is of class cr- l • It is easy to see,
õt
by using properties of flows, that its integral curves are the maps
1.9. PROPOSITION. The set of all flows generated by the vector field
X, ordered by inclusion, has a unique maximal element.
The "union" ·of all flows generated by X is indeed itself a
flow generated by X.